OFFSET
0,2
COMMENTS
A k-packed matrix of size n X n is a matrix with entries in the alphabet A_k = {0,1, ..., k} such that each row and each column contains at least one nonzero entry.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..100
H. Cheballah, S. Giraudo, R. Maurice, Combinatorial Hopf algebra structure on packed square matrices, arXiv preprint arXiv:1306.6605 [math.CO], 2013.
FORMULA
Cheballah et al. give an explicit formula.
From Andrew Howroyd, Sep 20 2017: (Start)
a(n) = Sum_{r=1..n} Sum_{i=0..r} Sum_{j=0..r} (-1)^(i+j) * binomial(r,i) * binomial(r,j) * binomial(i*j,n) * 2^n.
a(n) = 2^n * A104602(n).
(End)
MATHEMATICA
b[n_] := Sum[StirlingS1[n, k]*Sum[(m!)^2*StirlingS2[k, m]^2, {m, 0, k}], {k, 0, n}]/n!;
a[n_] := 2^n*b[n];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)
PROG
(PARI) \\ here b(n) is A104602.
b(n) = {sum(m=0, n, sum(k=0, n, stirling(n, k, 1) * m!^2 * stirling(k, m, 2)^2)) / n!}
a(n) = 2^n * b(n); \\ Andrew Howroyd, Sep 20 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Nov 09 2013
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Sep 20 2017
STATUS
approved